# Ornstein Zernike Equation

Ornstein-Zernike (OZ) equation is an integral equation used for calculating **correlation functions of two molecules especially in fluids**. By solving OZ equations we can prescribe the **distributions of molecule** as a function of position and time.

## Derivations
### General way
Let us define functions $h$:
\begin{eqnarray}
h(r_{12}) = g(r_{12}) - 1
\end{eqnarray}
which is a measure of influence of two molecules 1 and 2, where $h(r)$ is radial distributuion function. Ornstein and Zernike proposed $h(r_{12})$ as
\begin{eqnarray}
h(r_{12}) = c(r_{12}) + \rho \int_{}^{}c(r_{13}) h(r_{13})d\vec r_{3} \tag{1}
\end{eqnarray}
which is called the Ornstein-Zernike equation. The first term of RHS is the direct correlation function of the two molecules. The second term is the indirect term, which involves effects of other molecule labeled as 3.

#### Convolute form
The convolute form of equation (1) is
\begin{eqnarray}
h = c + \rho c h
\end{eqnarray}
which yeilds
\begin{eqnarray}
h = \frac{c}{1-\rho c}, ~~ c = \frac{h}{1+\rho h}.
\end{eqnarray}

To solve this equation one need another equation namely closure relation. The exmalples of closure relation is Percus–Yevick approximation that is useful when hard sphere potentials, and hypernetted-chain equation for soft potentials.

> closure means completeness (for being able to solve equation)

## memo
In "相転移・臨海現象と繰り込み群(p.84)" by Takahashi et al, OZ equation is derived by Landau theory. Are there any relevance?
### Landau Theory